Nonequispaced fast Fourier transforms for bandlimited functions

TL;DR

This paper introduces a new nonequispaced fast Fourier transform (NFFT)-like method for efficiently approximating bandlimited functions at arbitrary points using regularized Shannon sampling formulas, extending existing techniques for trigonometric polynomials.

Contribution

The paper develops a novel NFFT-like algorithm specifically designed for bandlimited functions, based on regularized Shannon sampling formulas, enhancing approximation capabilities.

Findings

The new method efficiently approximates bandlimited functions at nonequispaced points.

It extends NFFT techniques from trigonometric polynomials to general bandlimited functions.

The approach improves accuracy and computational efficiency for function evaluation problems.

Abstract

In this paper we consider the problem of approximating function evaluations $f(\boldsymbol x_j)$ at given nonequispaced points $\boldsymbol x_j$, $j=1,\dots N$, of a bandlimited function from given values $\hat{f}(\boldsymbol k)$, $\boldsymbol k\in \mathcal I_{\boldsymbol M}$, of its Fourier transform. Note that if a trigonometric polynomial is given, it is already known that this problem can be solved by means of the nonequispaced fast Fourier transform (NFFT). In other words, we introduce a new NFFT-like procedure for bandlimited functions, which is based on regularized Shannon sampling formulas.